\begin{align*}{\equiv} = \bigcap_{U \in \mathcal{U}}U.\end{align*}
\begin{align*}\eta(x) = \bigcap_{U \in \mathcal{U}} U(x).\end{align*}
\begin{align*}U_x = \smashoperator[r]{\bigcap_{\substack{U \\ x \in U}}} \enspace U = \smashoperator[r]{\bigcap_{U \in \mathcal N(x)}} \enspace U,\end{align*}
\begin{align*}\bigcap_{\substack{U \in \tau \\ x \in U}} \eta^\rightarrow(U) = \smashoperator[r]{\bigcap_{\substack{V \in \tau_{\equiv} \\ \eta(x) \in V}}} \enspace V = U_{\eta(x)}.\end{align*}
\begin{align*}\eta^\rightarrow(U_x) = \eta^\rightarrow\left(\bigcap_{\substack{U \in \tau \\ x \in U}} U\right) \subseteq \bigcap_{\substack{U \in \tau \\ x \in U}} \eta^\rightarrow(U) = U_{\eta(x)}.\end{align*}
\begin{align*} U_{\pi(x)} = U_x.\end{align*}
\begin{align*}F(x,t) =\begin{cases}x& 0 \leq t < 1, \\\pi(x) & t = 1.\end{cases}\end{align*}
\begin{align*} V = V_1 \oplus \ldots \oplus V_k\end{align*}
\begin{align*} \mathfrak h^\perp = \mathfrak p_u \oplus (\mathfrak d^\perp \cap \mathfrak c) \oplus (\mathfrak h^\perp \cap \mathfrak p^+_u).\end{align*}
\begin{align*}\widetilde{a}_{\mu, \nu}=\sum_{\mu^{\prime}, \nu^{\prime}}a_{\mu^{\prime}, \nu^{\prime}} R(\mu^{\prime}- \mu, \nu^{\prime}- \nu). \end{align*}
\begin{align*} \mathfrak u = \bigoplus\limits_{\Omega \in \widetilde \Psi} \mathfrak u(\Omega)\end{align*}
\begin{align*} \mathfrak h^\perp = \mathfrak p_u \oplus (\mathfrak d^\perp \cap \mathfrak c) \oplus \bigoplus\limits_{\Omega \in \widetilde \Psi} \mathfrak u(\Omega).\end{align*}
\begin{align*} \mathfrak h_\infty^\perp = \mathfrak p_u \oplus (\mathfrak d^\perp \cap \mathfrak c) \oplus \bigoplus\limits_{\Omega \in \widetilde \Psi} \mathfrak u_\infty(\Omega)\end{align*}
\begin{align*} \mathfrak h^\perp = \mathfrak p_u \oplus \bigoplus \limits_{\mu \in \Psi} \mathfrak g(\mu).\end{align*}
\begin{align*} \mathfrak g = (h_\delta^\perp \cap \mathfrak t) \oplus \bigoplus \limits_{\alpha \in Y(\delta)} V(\alpha).\end{align*}
\begin{align*} \mathfrak h^\perp = \bigoplus \limits_{\alpha \in Y(\delta)} (\mathfrak h^\perp \cap V(\alpha)).\end{align*}
\begin{align*} \Psi = \Psi_1 \cup \ldots \cup \Psi_p\end{align*}
\begin{align*} \sum \limits_{\mu \in \Psi_i} a_\mu \widehat \mu_* + \rho_i = \sigma = \sum \limits_{\nu \in \Psi_j} b_\nu \widehat \nu_* + \rho_j\end{align*}
\begin{gather*}\Psi_i^+ = \lbrace \mu \in \Psi_i \mid a_\mu > 0 \rbrace, \Psi_i^- = \lbrace \mu \in \Psi_i \mid a_\mu < 0 \rbrace, \\\Psi_j^+ = \lbrace \nu \in \Psi_j \mid b_\nu > 0 \rbrace, \Psi_j^- = \lbrace \nu \in \Psi_j \mid b_\nu < 0 \rbrace\end{gather*}
\begin{align*} \Sigma_G(G/N_1) = \lbrace \alpha_2, \alpha_3 \rbrace \ \ \Sigma_G(G/N_2) = \lbrace \alpha_1, \alpha_2 \rbrace.\end{align*}
\begin{align*}R(\alpha, \beta) = \begin{cases}1 & \;\; (\alpha, \beta)=(0,0)\\0 & \;\; (\alpha, \beta)\neq (0,0). \end{cases}\end{align*}
\begin{align*}\bigcup_{\{j:1\leq j\leq m,\,j\ne i\}}D(B_i,B_j)=\lambda_i \boxtimes (G\backslash \{0\}),\end{align*}
\begin{align*}\bigcup_{\{i\,:\,|B_i|=w_t\}}\bigcup_{\{j:1\leq j\leq m,\,j\ne i\}}D(B_i,B_j)=\lambda_t \boxtimes (G\backslash \{0\}),\end{align*}
\begin{align*}\left|\bigcup\limits_{1\leq i\ne j\leq m}D(B_i,\widetilde{B}_j)\right|=\sum\limits_{1\leq i\leq m}\sum\limits_{1\leq j\leq m\atop j\ne i}\sum_{b\in B_i}|D(\{b\},\widetilde{B}_j)|=\sum\limits_{1\leq i\leq m}\sum\limits_{1\leq j\leq m\atop j\ne i}\sum_{b\in B_i}\widetilde{k}=\widetilde{k}a(m-1).\end{align*}
\begin{align*}\rho_{(n,m,a)}=\min_{K'}\left\{\frac{\lambda'}{\widetilde{k'}m}\,:\,\exists\, (n,m,K',a,\lambda')\, s.t.\, \sum_{1\leq i\leq m}k'_i=a\right\}.\end{align*}
\begin{align*}D^{2}_i=\{\alpha^{i+2j}\,:\,0\leq j\leq 2k-1\},\,\,\,\,i=0,1\end{align*}
\begin{align*}D^{4}_i=\{\alpha^{i+4j}\,:\,0\leq j\leq k-1\},\,\,\,\,0\leq i\leq 3.\end{align*}
\begin{align*}\left[ \widehat{q}_1,\widehat{p}_1 \right]=\left[ \widehat{q}_2,\widehat{p}_2 \right]=i,\end{align*}
\begin{align*}\left[ \widehat{q}_1,\widehat{p}_1 \right]=\left[ \widehat{q}_2,\widehat{p}_2 \right]=i, \left[ \widehat{q}_1,\widehat{q}_2 \right]= i \theta, \left[ \widehat{p}_1,\widehat{p}_2 \right]= i \eta,\end{align*}
\begin{align*}\left(\widehat{p}_1^2-\widehat{p}_2^2 -48 e^{-2 \sqrt{3}\widehat{q}_2}\right) \psi=0.\end{align*}
\begin{align*}\widehat{q}_1=x_1, \widehat{q}_2=x_2 , \widehat{p}_1= -i \frac{\partial}{\partial x_1} , \widehat{p}_2= -i \frac{\partial}{\partial x_2},\end{align*}
\begin{align*}\widetilde{\phi} (x) = \phi (x) + \phi (x-1). \end{align*}
\begin{align*}\left(\frac{\partial^2}{\partial x_2^2} - \frac{\partial^2}{\partial x_1^2} -48 e^{-2 \sqrt{3} x_2} \right) \psi(x_1,x_2)=0,\end{align*}
\begin{align*}\psi(x_1,x_2) =\psi_{\nu}^{\pm} (x_1,x_2)=e^{\pm i \nu \sqrt{3}x_1}K_{i \nu}\left(4e^{- \sqrt{3}x_2}\right),\end{align*}
\begin{align*}\begin{array}{c}\left\{ \left(i \mu \frac{\partial}{\partial x_1} + \frac{\eta x_2}{2 \mu} \right)^2 -\left(i \mu \frac{\partial}{\partial x_2} - \frac{\eta x_1}{2 \mu} \right)^2\right.\\ \\ \left. -48 \exp \left[ -2 \sqrt{3} \left(\lambda x_2 + \frac{i \theta}{2 \lambda}\frac{\partial}{\partial x_1} \right) \right]\right\} \psi(x_1,x_2)=0\end{array}\end{align*}
\begin{align*} \psi_a(x_1,x_2)= \mathcal{R}_a (x_2)\exp \left[ \frac{i x_1}{\mu} \left(a-\frac{\eta}{2 \mu} x_2 \right) \right],\end{align*}
\begin{align*}\begin{array}{c}V(x)=-(\eta x-a)^2-F^2\mu^4x^4-2 F \mu^2 (\eta x-a)x^2+\\\\+48 \exp \left(-2 \sqrt{3} x-2 \sqrt{3} \mu^2Ex^2+ \frac{\sqrt{3} \theta a}{\mu \lambda} \right),\end{array}\end{align*}
\begin{align*}\widetilde{f} (\xi) := (2 \pi )^{- d/2} \int_{\mathbb{R}^d} f(x) e^{- i x \cdot \xi} dx.\end{align*}
\begin{align*}\mathcal{F} f( \omega) := \int_{\mathbb{R}^d} f(t) e^{-2 \pi i t \cdot \omega} dt\end{align*}
\begin{align*}E(\mu)=-\int_{\mathbb{R}} \mu(x) \log \left(\mu(x) \right) dx,\end{align*}
\begin{align*}(\widehat{A} -a \widehat{I}) f = i c (\widehat{B} -b \widehat{I}) f\end{align*}
\begin{align*}\begin{array}{l}\Delta_x (f,a)= \|(\widehat{x} -a \widehat{I}) f \|_{L^2 (\mathbb{R})} = \left(\int_{\mathbb{R}} (x-a)^2 | f (x)|^2 dx \right)^{\frac{1}{2}}\\\\\Delta_{\xi} (f,b)= \|(\widehat{\xi} -b \widehat{I}) f \|_{L^2 (\mathbb{R})} = \left(\int_{\mathbb{R}} (\xi-b)^2 | \widetilde{f} (\xi)|^2 d \xi . \right)^{\frac{1}{2}}\end{array}\end{align*}
\begin{align*}\begin{aligned} = \!{\sigma Loc}_\sigma &\simeq \!{QPol} := , \\ = \!{\sigma BorLoc}_\sigma &\simeq \!{SBor} := .\end{aligned}\end{align*}
\begin{align*}\begin{array}{l}<x>_f= \langle \widehat{x}f, f \rangle_{L^2 (\mathbb{R})}= \int_{\mathbb{R}} x | f (x)|^2 dx \\\\<\xi>_f= \langle \widehat{\xi}f, f \rangle_{L^2 (\mathbb{R})}= \int_{\mathbb{R}} \xi | \widetilde{f} (\xi)|^2 d \xi .\end{array}\end{align*}
\begin{align*}D_s f (x) = \frac{1}{\sqrt{|s|}} f \left( \frac{x}{s} \right)\end{align*}
\begin{align*}\widetilde{D_s f} (\xi) = D_{\frac{1}{s}} \widetilde{f} (\xi)\end{align*}
\begin{align*}\Delta_A (f_1,0) \Delta_B (f_1,0) = C_1\end{align*}
\begin{align*}\begin{array}{c}\Delta_A (f_s,0)= \left(\int_{\mathbb{R}} x^{2n} | f_s (x)|^2 dx\right)^{\frac{1}{2}} = |s|^{-1/2} \left(\int_{\mathbb{R}} x^{2n} | f_1 \left( s^{-1} x\right)|^2 dx \right)^{\frac{1}{2}}=\\\\=|s|^{-1/2} \left(\int_{\mathbb{R}} (sy)^{2n} | f_1 \left(y\right)|^2 |s|dy \right)^{\frac{1}{2}}= |s|^n \Delta_A (f_1,0)\end{array}\end{align*}
\begin{align*}\Delta_B (f_s,0) = |s|^{-m} \Delta_B (f_1,0)\end{align*}
\begin{align*}\Delta_A (f_s,0) \Delta_B (f_s,0) = |s|^{n-m} C_1\end{align*}
\begin{align*}|s|^{n-m} C_1=\Delta_A (f_s,0) \Delta_B (f_s,0) \geq \Delta_A (f_s,<A>_{f_s}) \Delta_B (f_s,<B>_{f_s}) \to 0,\end{align*}
\begin{align*}\langle \left[(\widehat x)^{2k}, (\widehat \xi)^{2k} \right] f , f \rangle_{L^2 (\mathbb{R})} = (-i )^{2k} \sqrt{\frac{2a}{\pi}} \int_{\mathbb{R}} e^{-a x^2} \left( x^{2k} \frac{d^{2k}}{dx^{2k}} - \frac{d^{2k}}{dx^{2k}} x^{2k} \right) e^{-a x^2} dx\end{align*}
\begin{align*}\| ~(x-a)^2 f \|_{L^2 (\mathbb{R})}= \left(\int_{\mathbb{R}} (x-a)^4 |f(x)|^2 dx \right)^{1/2},\end{align*}
\begin{align*}a \wedge \bigvee_i b_i = \bigvee_i (a \wedge b_i)\end{align*}
\begin{align*}\Delta_A (f,a^2) =\| ~(x^2-a^2) f \|_{L^2 (\mathbb{R})} = \left(\int_{\mathbb{R}} (x^2-a^2)^2 |f(x)|^2 dx \right)^{1/2}.\end{align*}
\begin{align*}\|\widehat{A} f \|_{L^2 (\mathbb{R})} = \|\widehat{B} f\|_{L^2 (\mathbb{R})}\end{align*}
\begin{align*} \widehat{R} = \epsilon \left( \widehat{q}_1 + \frac{\theta}{1+ \sqrt{1- \xi}} \widehat{p}_2 \right)\end{align*}
\begin{align*}\left[A, B \right]_{\rho} =\left[A, B \right]_0+ \sum_{k=1}^{\infty} B_k (A,B) \rho^k,\end{align*}
\begin{align*}\mathcal{B}^{\alpha} (\mathbb{R}^2) := \left\{f \in \mathcal{S}^{\prime} (\mathbb{R}^2): ~ \|f \|_{\alpha} < + \infty \right\}\end{align*}
\begin{align*}\|f \|_{\alpha}^2 := 2 \|f \|_{L^2 (\mathbb{R}^2)} + \|\widehat{u} f\|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{v} f\|_{L^2 (\mathbb{R}^2)}^2\end{align*}
\begin{align*}\mathcal{B} (\mathbb{R}^2) := \left\{f \in \mathcal{S}^{\prime} (\mathbb{R}^2): ~ \|f \|_{\mathcal{B}} < + \infty \right\}\end{align*}
\begin{align*}\langle f,g \rangle_{\alpha} : = 2 \langle f,g \rangle_{L^2 (\mathbb{R}^2)} + \langle \widehat{u}f,\widehat{u}g \rangle_{L^2 (\mathbb{R}^2)} + \langle \widehat{v}f,\widehat{v}g \rangle_{L^2 (\mathbb{R}^2)}\end{align*}
\begin{align*}\|f_n-f_m\|_{\alpha} = \left( 2\|f_n-f_m\|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{q_1}(f_n-f_m)\|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{q_2}(f_n-f_m)\|_{L^2 (\mathbb{R}^2)}^2 \right)^{1/2} < \epsilon\end{align*}
\begin{align*}\|\widehat{q_2}f\|_{L^2 (\mathbb{R}^2)}^2 = \lambda^2 \|x_2 f\|_{L^2 (\mathbb{R}^2)}^2 + \frac{\theta^2}{4 \lambda^2} \| \xi_1 \widetilde{f}\|_{L^2 (\mathbb{R}^2)}^2.\end{align*}
\begin{align*}B_\kappa := \{b \in B \mid b \}.\end{align*}
\begin{align*}\|\widehat{u} f \|_{L^2 (\mathbb{R}^2)} \leq \left( 2 \| f \|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{u} f \|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{v} f\|_{L^2 (\mathbb{R}^2)}^2\right)^{\frac{1}{2}} = \| f \|_{\alpha}\end{align*}
\begin{align*}S:= \left\{f \in \mathcal{B}^{\alpha} (\mathbb{R}^2): ~ \| f \|_{L^2 (\mathbb{R}^2)} =1 \right\}.\end{align*}
\begin{align*}\overline{B}_R^{(\alpha)}:= \left\{f \in \mathcal{B}^{\alpha} (\mathbb{R}^2): ~\| f\|_{\alpha} \leq R \right\}.\end{align*}
\begin{align*}U_R^{(\alpha)}= \left\{ f \in \overline{B}_R^{(\alpha)}: ~ \|f \|_{L^2(\mathbb{R}^2)} =1 \right\}\end{align*}
\begin{align*}\langle u,v\rangle_{L^2 (\mathbb{R}^2)}=\langle\widehat{A} u,v\rangle_{\alpha},\end{align*}
\begin{align*}\langle h_l-g,u\rangle_{L^2 (\mathbb{R}^2)}=\langle\widehat{A} ( h_l-g), u\rangle_{\alpha}.\end{align*}
\begin{align*}f \in S:= \left\{f \in \mathcal{B}^{\alpha} (\mathbb{R}^2): ~ \|f\|_{L^2 (\mathbb{R}^2)} =1 \right\}.\end{align*}
\begin{align*}F^{(\alpha)} \left[f \right]= \| f \|_{\alpha}^2 -2 \|f \|_{L^2 (\mathbb{R}^2)}^2 =\| f \|_{\alpha}^2 -2 ,\end{align*}
\begin{align*}\|f \|_{L^2 (\mathbb{R}^2)} = 1.\end{align*}
\begin{align*}\mathfrak{L}^{(\alpha)} \left[f , \gamma \right] = F^{(\alpha)} \left[f \right] + \gamma \left( 1- \|f \|_{L^2 (\mathbb{R}^2)}^2 \right),\end{align*}
\begin{align*}A_\top := A \sqcup \{\top\}\end{align*}
\begin{align*}\langle f, g \rangle_{L^2 (\mathbb{R}^2)}= \langle \widehat{A}f,g \rangle_{\alpha},\end{align*}
\begin{align*}\langle \widehat{A}f, g \rangle_{\alpha} = \langle f, g \rangle_{L^2 (\mathbb{R}^2)} = \overline{\langle g,f \rangle}_{L^2 (\mathbb{R}^2)} = \overline{\langle \widehat{A}g,f \rangle }_{\alpha} = \langle f,\widehat{A}g \rangle_{\alpha} .\end{align*}
\begin{align*}\langle \widehat{A} f,f \rangle_{\alpha} = \|f \|_{L^2 (\mathbb{R}^2)}^2 >0,\end{align*}
\begin{align*}0= \langle \widehat{A}g,f \rangle_{\alpha} = \langle g,f \rangle_{L^2 (\mathbb{R}^2)},\end{align*}
\begin{align*} F^{(\alpha)} [f_0] = _{f \in S} F^{(\alpha)} \left[f \right].\end{align*}
\begin{align*}\langle \widehat{u}g, \widehat{u}f_0 \rangle_{L^2 (\mathbb{R}^2)} + \langle \widehat{v}g, \widehat{v}f_0 \rangle_{L^2 (\mathbb{R}^2)} = F^{(\alpha)}[f_0] \langle g,f_0 \rangle_{L^2 (\mathbb{R}^2)},\end{align*}
\begin{align*}\langle \widehat{u}g, \widehat{u}f_0 \rangle_{L^2 (\mathbb{R}^2)} + \langle \widehat{v}g, \widehat{v}f_0 \rangle_{L^2 (\mathbb{R}^2)} = \gamma \langle g,f_0 \rangle_{L^2 (\mathbb{R}^2)},\end{align*}
\begin{align*}\left(\widehat{u}^2 + \widehat{v}^2 \right) f_0= \gamma f_0.\end{align*}
\begin{align*}\widehat{H}^{(\alpha)} f_{\nu} = \nu f_{\nu}\end{align*}
\begin{align*}\|\widehat{u} f_{\nu} \|_{L^2 (\mathbb{R}^2)}^2 + \|\widehat{v} f_{\nu} \|_{L^2 (\mathbb{R}^2)}^2 = \nu \| f_{\nu} \|_{L^2 (\mathbb{R}^2)}^2\end{align*}
\begin{align*} \begin{cases} \lim_{t\to 0}\mathbf{F}(t,t)=1, &\\ \lim_{t\to 0}\mathbf{F}(s,t)=0, & \end{cases} \\ \mathbf{F}(\cdot,t)\ \text{is decreasing, for every fixed $t>0$;}\\ \ \mathbf{F}(s,t)\ \text{is uniformly continuous on $s\geq t\geq \tau$.}\end{align*}
\begin{align*}a \wedge \neg a = \bot, &&a \vee \neg a = \top\end{align*}
\begin{align*}\nu = F^{(\alpha)} \left[ f_{\nu} \right]\end{align*}
\begin{align*}\nu = F^{(\alpha)} \left[ f_{\nu} \right] \geq F^{(\alpha)} \left[f_0 \right]\end{align*}
\begin{align*}\langle g,f_0 \rangle_{\alpha} = (\gamma+2) \langle g, f_0 \rangle_{L^2 (\mathbb{R}^2)}.\end{align*}
\begin{align*}\langle g,f_0 \rangle_{\alpha} = (\gamma +2) \langle \widehat{A} g, f_0 \rangle_{ \alpha} \Leftrightarrow \langle g,f_0 \rangle_{\alpha} = (\gamma+2) \langle g, \widehat{A} f_0 \rangle_{ \alpha},\end{align*}
\begin{align*}f_0 =(\gamma +2) \widehat{A} f_0 \Leftrightarrow \left(\widehat{A}^{-1} - 2 \widehat{I} \right)f_0= \gamma f_0.\end{align*}
\begin{align*}f (x_1, x_2) = \left(\frac{4}{\pi^2 ab} \right)^{\frac{1}{4}} \exp \left[ - \frac{1}{a} (x_1 -x_1^{(0)})^2 - \frac{1}{b} (x_2 - x_2^{(0)})^2 \right]\end{align*}
\begin{align*}x_1^{(0)} = - \frac{\lambda}{2E},\end{align*}
\begin{align*}\begin{array}{c}\langle \widehat{q}_1f, f \rangle_{L^2 (\mathbb{R}^2)}= \int_{\mathbb{R}^2} \left[\left(\lambda x_1 + \frac{i \theta}{2 \lambda} \frac{\partial}{\partial x_2} + E x_1^2 \right)f (x_1, x_2) \right] \overline{f (x_1, x_2)} dx_1 dx_2 =\\\\= \lambda x_1^{(0)} + E \left((x_1^{(0)})^2 + \frac{a}{4} \right)\end{array}\end{align*}
\begin{align*}\begin{array}{c}\langle \widehat{q}_1^2 f,f \rangle_{L^2 (\mathbb{R}^2)}=E^2 (x_1^{(0)})^4 + 2 \lambda E (x_1^{(0)})^3 + \left(\lambda^2 + \frac{3}{2} a E^2 \right) (x_1^{(0)})^2 + \frac{3}{2} \lambda a E x_1^{(0)} + \\\\+\frac{\lambda^2 a}{4} + \frac{\theta^2}{4 \lambda^2 b} + \frac{3 a^2 E^2}{16}\end{array}\end{align*}
\begin{align*}\Delta_{q_1} (f, <q_1>_f) = \left[ a \left( E x_1^{(0)} + \frac{\lambda}{2} \right)^2 + \frac{\theta^2}{4 \lambda^2 b} + \frac{a^2 E^2}{8} \right]^{\frac{1}{2}}\end{align*}
\begin{align*}a \wedge \neg a = \bot, &&a \vee \neg a = \top_{A_i}\end{align*}
\begin{align*}\Delta_{q_1} (f, <q_1>_f) =\left( \frac{\theta^2}{4 \lambda^2 b} + \frac{a^2 E^2}{8} \right)^{\frac{1}{2}}\end{align*}
\begin{align*}\langle \widehat{p}_1 f ,f \rangle_{L^2 (\mathbb{R}^2)}= \int_{\mathbb{R}^2} \left[\left(-i \mu \frac{\partial}{\partial x_1} + \frac{\eta}{2 \mu} x_2 \right)f (x_1, x_2) \right] \overline{f (x_1, x_2)} dx_1 dx_2 = \frac{\eta}{2 \mu} x_2^{(0)}\end{align*}
\begin{align*}\langle \widehat{p}_1^2 f ,f \rangle_{L^2 (\mathbb{R}^2)}= \frac{\mu^2}{a} + \left(\frac{\eta}{2 \mu} \right)^2 \left((x_2^{(0)})^2 + \frac{b}{4} \right)\end{align*}
\begin{align*}\Delta_{p_1} (f, <p_1>_f) = \frac{\mu}{\sqrt{a}} \left[1 + \left(\frac{\eta}{4 \mu^2} \right)^2 ab \right]^{\frac{1}{2}}\end{align*}
\begin{align*}\Delta_{q_1} (f, <q_1>_f) \Delta_{p_1} (f, <p_1>_f) = \frac{1}{2} \left( \frac{1}{2} \mu^2 E^2 a + \frac{\mu^2 \theta^2}{\lambda^2 ab} + \frac{\eta^2 E^2 a^2 b}{32 \mu^2} + \frac{\theta^2 \eta^2}{16 \mu^2 \lambda^2} \right)^{\frac{1}{2}}\end{align*}
\begin{align*}\Delta_{q_1} (f, <q_1>_f) \Delta_{p_1} (f, <p_1>_f) \to \frac{\theta \eta}{8 \mu \lambda} = \frac{\xi}{4 (1 + \sqrt{1- \xi})} \leq \frac{\xi}{4} < \frac{1}{2}\end{align*}
\begin{align*} V_gf(x,\omega)=\langle f,\pi(x,\omega)g \rangle_{L^2 (\mathbb{R}^d)}=\int_{\mathbb{R}^d} f(t)\overline{g(t-x)}e^{-2\pi it \cdot \omega}dt.\end{align*}
\begin{align*}V_gf(x,\omega)=\langle f,\pi(x,\omega) \overline{g} \rangle.\end{align*}
\begin{align*}\|F\|_{L_{x,\omega}^{r,s} (\mathbb{R}^{2d})}=\left(\int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d} |F(x, \omega)|^r d x\right)^{\frac{s}{r}} d \omega\right)^{\frac{1}{s}},\end{align*}
\begin{align*}\|f\|_{M_m^{r,s} (\mathbb{R}^d)}= \|m V_gf\|_{L_{x,\omega}^{r,s} (\mathbb{R}^{2d})} = \left(\int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d} |V_g f(x, \omega) m (x, \omega) |^r d x\right)^{\frac{s}{r}} d \omega\right)^{\frac{1}{s}} < \infty.\end{align*}
\begin{align*}\top_{A_i} \wedge \top_{A_j} = \bot \quad.\end{align*}
\begin{align*}\langle V_{g_1} f_1, V_{g_2} f_2 \rangle_{L^2 (\mathbb{R}^{2d})} = \langle f_1,f_2 \rangle_{L^2 (\mathbb{R}^d)} \overline{\langle g_1,g_2 \rangle}_{L^2 (\mathbb{R}^d)},\end{align*}
\begin{align*}\lim_{R \to \infty} \left( \mbox{ess~sup}_{|z| \ge R} |f(z)| \right)=0 ,\end{align*}
\begin{align*}m(x,\omega)= \sqrt{| \psi(x)|^2 + |\phi (\omega)|^2}.\end{align*}
\begin{align*}\mathcal{B} (\mathbb{R}^2)=M_m^2 (\mathbb{R}^2).\end{align*}